Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic

Abstract

We prove Todd's relation conjecture: all ${\mathbb{F}}_q(\theta )$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators ${\mathcal{B}}^{\mathrm{S}}$, ${\mathcal{C}}^{\mathrm{S}}$, ${\mathcal{B}}^{\mathrm{S}}\circ {\mathcal{C}}^{\mathrm{S}}$; moreover they are also generated by ${\mathcal{B}}^{\ast \mathrm{S}}$, ${\mathcal{C}}^{\mathrm{S}}$, ${\mathcal{B}}^{\ast \mathrm{S}}\circ {\mathcal{C}}^{\mathrm{S}}$. The ${\mathcal{B}}^{\ast \mathrm{S}}$-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the ${\mathcal{B}}^{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let ${\mathfrak{BR}}_w^{\mathrm{S}}$ be the ${\mathbb{F}}_q(\theta )$-linear space spanned by binary relations of weight $w$, and let ${\mathrm{Fix}}_w^{\mathrm{S}}$ be the ${\mathbb{F}}_q(\theta )$-linear space spanned by fixed relations. We derive generating functions ${\sum }_{w\ge 1}\left({\dim }_{{\mathbb{F}}_q(\theta )}{\mathrm{Fix}}_w^{\mathrm{S}}\right)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$ and ${\sum }_{w\ge 1}\left({\dim }_{{\mathbb{F}}_q(\theta )}{\mathfrak{BR}}_w^{\mathrm{S}}\right)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$ Our results are based on the recent work of Im-Kim-Ngo Dac on the ${\mathbb{F}}_q$-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.

17 pages